3.254 \(\int \frac{\sinh ^7(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\)
Optimal. Leaf size=290 \[ \frac{3 \left (\sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} b^{7/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{3 \left (\sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} b^{7/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{32 b d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]
[Out]
(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*
Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*b*d*(a - b +
2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (Cosh[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cosh[c + d*x]^2))/(32*
(a - b)^2*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))
________________________________________________________________________________________
Rubi [A] time = 0.467115, antiderivative size = 290, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{3215, 1205, 1178, 1166, 205, 208} \[ \frac{3 \left (\sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} b^{7/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{3 \left (\sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} b^{7/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{32 b d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]
[Out]
(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*
Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*b*d*(a - b +
2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (Cosh[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cosh[c + d*x]^2))/(32*
(a - b)^2*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))
Rule 3215
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rule 1205
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(
2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Rule 1178
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{4 a (a-4 b)-2 a (3 a-8 b) x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a (a-b) b d}\\ &=-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac{\cosh (c+d x) \left (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-12 a^2 (a-5 b) b+12 a^2 (a-3 b) b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac{\cosh (c+d x) \left (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac{\left (3 \left (\sqrt{a}+2 \sqrt{b}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^2 b d}-\frac{\left (3 \left (a^{3/2}-3 \sqrt{a} b-2 b^{3/2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt{a} (a-b)^2 b d}\\ &=\frac{3 \left (\sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} b^{7/4} d}-\frac{3 \left (\sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} b^{7/4} d}-\frac{a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac{\cosh (c+d x) \left (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 1.29495, size = 802, normalized size = 2.77 \[ \frac{-\frac{32 \cosh (c+d x) (-7 a+25 b+3 (a-3 b) \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}-3 \text{RootSum}\left [b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\& ,\frac{-a c \text{$\#$1}^6+3 b c \text{$\#$1}^6-a d x \text{$\#$1}^6+3 b d x \text{$\#$1}^6-2 a \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^6+6 b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^6+3 a c \text{$\#$1}^4-17 b c \text{$\#$1}^4+3 a d x \text{$\#$1}^4-17 b d x \text{$\#$1}^4+6 a \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^4-34 b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^4-3 a c \text{$\#$1}^2+17 b c \text{$\#$1}^2-3 a d x \text{$\#$1}^2+17 b d x \text{$\#$1}^2-6 a \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^2+34 b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right ) \text{$\#$1}^2+a c-3 b c+a d x-3 b d x+2 a \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right )-6 b \log \left (\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right ) \text{$\#$1}\right )}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\& \right ]+\frac{512 a (a-b) (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{256 (a-b)^2 b d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]
[Out]
((-32*Cosh[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh
[4*(c + d*x)]) + (512*a*(a - b)*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] +
b*Cosh[4*(c + d*x)])^2 - 3*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (a*c - 3*b*c +
a*d*x - 3*b*d*x + 2*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1
] - 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 3*a*c*#1^2
+ 17*b*c*#1^2 - 3*a*d*x*#1^2 + 17*b*d*x*#1^2 - 6*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x
)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 34*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a*c*#1^4 - 17*b*c*#1^4 + 3*a*d*x*#1^4 - 17*b*d*x*#1^4 + 6*a*Log[-Cosh[(c + d*
x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 34*b*Log[-Cosh[(c + d*x)/2] -
Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - a*c*#1^6 + 3*b*c*#1^6 - a*d*x*#1^6 + 3
*b*d*x*#1^6 - 2*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1
^6 + 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*
#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)
________________________________________________________________________________________
Maple [B] time = 0.088, size = 2563, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x)
[Out]
3/64/d/b^2/(a^2-2*a*b+b^2)*a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2
*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+1/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*
d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/
2*c)^2-32/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2
*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8-3/8/d/(tanh(1/2*d*x+1/2*c)^8*
a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^
2*a^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-9/64/d/b/(a^2-2*a*b+b^2)/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*
(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)-9/64/d/b/(a^2-2*a*b+b^2)
/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/
2))*(a*b)^(1/2)-35/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(
1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*a^2+5/d/(tanh(1/2*d*x+
1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c
)^2*a+a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*a^2-25/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6
*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)*t
anh(1/2*d*x+1/2*c)^4+2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tan
h(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*a^2+3/64/d/b^2/(a^2
-2*a*b+b^2)*a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^
(1/2)*a)^(1/2))*(a*b)^(1/2)+8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-1
6*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8-42/d/(tanh(1/
2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x
+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-1/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^
6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)+
13/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*a+95/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh
(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2
*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*a-27/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2
*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+19/8
/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*t
anh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-3/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*
d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*
b+b^2)*tanh(1/2*d*x+1/2*c)^14+15/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^
4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-111/8/d
/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tan
h(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*a+10/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*
x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+
b^2)*tanh(1/2*d*x+1/2*c)^10-1/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a
-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)+8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*ta
nh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a
^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+3/32/d/(a^2-2*a*b+b^2)/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*
d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))-3/32/d/(a^2-2*a*b+b^2)/(-a*b+(a*b)^(1/2)*a)^(1/2
)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
1/16*(3*(a*b*e^(15*c) - 3*b^2*e^(15*c))*e^(15*d*x) - (23*a*b*e^(13*c) - 77*b^2*e^(13*c))*e^(13*d*x) + (16*a^2*
e^(11*c) + 131*a*b*e^(11*c) - 177*b^2*e^(11*c))*e^(11*d*x) - (144*a^2*e^(9*c) + 367*a*b*e^(9*c) - 109*b^2*e^(9
*c))*e^(9*d*x) - (144*a^2*e^(7*c) + 367*a*b*e^(7*c) - 109*b^2*e^(7*c))*e^(7*d*x) + (16*a^2*e^(5*c) + 131*a*b*e
^(5*c) - 177*b^2*e^(5*c))*e^(5*d*x) - (23*a*b*e^(3*c) - 77*b^2*e^(3*c))*e^(3*d*x) + 3*(a*b*e^c - 3*b^2*e^c)*e^
(d*x))/(a^2*b^3*d - 2*a*b^4*d + b^5*d + (a^2*b^3*d*e^(16*c) - 2*a*b^4*d*e^(16*c) + b^5*d*e^(16*c))*e^(16*d*x)
- 8*(a^2*b^3*d*e^(14*c) - 2*a*b^4*d*e^(14*c) + b^5*d*e^(14*c))*e^(14*d*x) - 4*(8*a^3*b^2*d*e^(12*c) - 23*a^2*b
^3*d*e^(12*c) + 22*a*b^4*d*e^(12*c) - 7*b^5*d*e^(12*c))*e^(12*d*x) + 8*(16*a^3*b^2*d*e^(10*c) - 39*a^2*b^3*d*e
^(10*c) + 30*a*b^4*d*e^(10*c) - 7*b^5*d*e^(10*c))*e^(10*d*x) + 2*(128*a^4*b*d*e^(8*c) - 352*a^3*b^2*d*e^(8*c)
+ 355*a^2*b^3*d*e^(8*c) - 166*a*b^4*d*e^(8*c) + 35*b^5*d*e^(8*c))*e^(8*d*x) + 8*(16*a^3*b^2*d*e^(6*c) - 39*a^2
*b^3*d*e^(6*c) + 30*a*b^4*d*e^(6*c) - 7*b^5*d*e^(6*c))*e^(6*d*x) - 4*(8*a^3*b^2*d*e^(4*c) - 23*a^2*b^3*d*e^(4*
c) + 22*a*b^4*d*e^(4*c) - 7*b^5*d*e^(4*c))*e^(4*d*x) - 8*(a^2*b^3*d*e^(2*c) - 2*a*b^4*d*e^(2*c) + b^5*d*e^(2*c
))*e^(2*d*x)) + 1/128*integrate(24*((a*e^(7*c) - 3*b*e^(7*c))*e^(7*d*x) - (3*a*e^(5*c) - 17*b*e^(5*c))*e^(5*d*
x) + (3*a*e^(3*c) - 17*b*e^(3*c))*e^(3*d*x) - (a*e^c - 3*b*e^c)*e^(d*x))/(a^2*b^2 - 2*a*b^3 + b^4 + (a^2*b^2*e
^(8*c) - 2*a*b^3*e^(8*c) + b^4*e^(8*c))*e^(8*d*x) - 4*(a^2*b^2*e^(6*c) - 2*a*b^3*e^(6*c) + b^4*e^(6*c))*e^(6*d
*x) - 2*(8*a^3*b*e^(4*c) - 19*a^2*b^2*e^(4*c) + 14*a*b^3*e^(4*c) - 3*b^4*e^(4*c))*e^(4*d*x) - 4*(a^2*b^2*e^(2*
c) - 2*a*b^3*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x)
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)**7/(a-b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError